The encoding of messages has an important role in information theory. Two basic problems entail representing messages efficiently and transmitting messages precisely. The former is called “source coding” and is related to data compression. The latter is called “channel coding” and is concerned with error correction. All information processing techniques are connected with these two problems.
Messages comprise sequences of various letters and usually the frequencies of the letters are not equal. The unequal frequencies imply a redundancy that enables the compression of the message. Classical source coding entails the coding of common letters as short sequences of code symbols (such as the binary digits {0, 1}) and uncommon letters as longer code sequences. Shannon's source coding theorem [C. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948)] gives the bounds on the degree a classical message can be compressed. For a source alphabet {A, B, . . . , Z} with given prior probabilities {P(A), P(B), . . . , P(Z)}, the minimum average length of the encoded message is given by the Shannon entropy:
                    H        =                  -                                    ∑                                                n                  =                  A                                ,                B                ,                …                                      ⁢                                          P                ⁡                                  (                  n                  )                                            ⁢                              log                2                            ⁢                                                P                  ⁡                                      (                    n                    )                                                  .                                                                        (        1        )            H takes its maximum value when all letters appear with equal probability. Then any compression is impossible.
The quantum domain, however, offers new possibilities [C. H. Bennett and D. P. Di Vincenzo, Nature (London) 404, 247 (2000)]. In particular, there is another kind of redundancy when the letters are conveyed by non-orthogonal quantum states, |ΨA>, |ΨB>, |ΨC>, . . . . with corresponding probabilities, PA, PB, PC. . . , Namely, compression is possible even if PA=PB=PC=. . . , in contrast to the classical case. Recently, Schumacher and Jozsa derived the quantum version of the source coding theorem [B. Schumacher, Phys. Rev. A 51, 2738 (1995), R. Jozsa and B. Schumacher, J. Mod. Opts. 41, 2343 (1994)]. The quantum noiseless coding theorem implies that, by coding the quantum message in blocks of K letters, KS({circumflex over (ρ)}) qubits are sufficient to encode each block in the limit K→∞ where S({circumflex over (ρ)}) is the von Neumann entropy of the density operator{circumflex over (ρ)}=ΣPn|ψn><ψn|,that is,
      S    ⁡          (              ρ        ^            )        =      -                  ∑        i            ⁢                          ⁢                        λ          i                ⁢                  log          2                ⁢                                  ⁢                  λ          i                    where {λi} is the set of eigenvalues of {circumflex over (ρ)}.
Quantum source coding plays a central role in quantum information theory. In addition, quantum source coding has practical advantages in the compression of non-orthogonal data sets. For example, in long-haul optical communication channels the optical signals suffer significant attenuation and one must deal with sequences of pulses in non-orthogonal coherent states. Compressing the sequences before storing or relaying to another channel can save expensive quantum channel resources. Sequences of non-orthogonal states are also essential for many quantum cryptographic schemes [N. Gisin, G. G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002)]. In particular, for a fixed rate of quantum key generation per transmitted letter state, any compression of the sequence will potentially give a more efficient use of the quantum channel.
Despite its importance from both fundamental and practical perspectives, quantum source coding has not, however, been demonstrated experimentally and, moreover, the specific apparatus able to realize it is unknown. One objective of the present invention is to provide an apparatus that can realize quantum source coding. Another objective of the present invention is to provide a quantum information communication system using the apparatus.